Model-based estimation of muscle and ACL forces during turning maneuvers in alpine skiing

In alpine skiing, estimation of the muscle forces and joint loads such as the forces in the ACL of the knee are essential to quantify the loading pattern of the skier during turning maneuvers. Since direct measurement of these forces is generally not feasible, non-invasive methods based on musculoskeletal modeling should be considered. In alpine skiing, however, muscle forces and ACL forces have not been analyzed during turning maneuvers due to the lack of three dimensional musculoskeletal models. In the present study, a three dimensional musculoskeletal skier model was successfully applied to track experimental data of a professional skier. During the turning maneuver, the primary activated muscles groups of the outside leg, bearing the highest loads, were the gluteus maximus, vastus lateralis as well as the medial and lateral hamstrings. The main function of these muscles was to generate the required hip extension and knee extension moments. The gluteus maximus was also the main contributor to the hip abduction moment when the hip was highly flexed. Furthermore, the lateral hamstrings and gluteus maximus contributed to the hip external rotation moment in addition to the quadratus femoris. Peak ACL forces reached 211 N on the outside leg with the main contribution in the frontal plane due to an external knee abduction moment. Sagittal plane contributions were low due to consistently high knee flexion (> 60\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document}∘), substantial co-activation of the hamstrings and the ground reaction force pushing the anteriorly inclined tibia backwards with respect to the femur. In conclusion, the present musculoskeletal simulation model provides a detailed insight into the loading of a skier during turning maneuvers that might be used to analyze appropriate training loads or injury risk factors such as the speed or turn radius of the skier, changes of the equipment or neuromuscular control parameters.

The contact between the skis and the snow was modeled using three force components acting on every ski segment and taking into account the side cut shape of the ski 1 . The three force components were a penetration force acting normal to the snow surface, a shear force acting parallel to the snow surface and orthogonal to the ski edge and finally a friction force.
Specifically, the penetration force F p was modeled as a function of the penetration depth e and penetration speedė of the ski edge orthogonal to the snow surface and the edging angle θ and incorporated a hypoplastic constitutive equation 1 . Following Mössner et al. 1 the penetration force F p is computed as follows assuming a ski segment is in contact with the snow surface (or setting e = max(e, 0)).
else (2) and f = 1 front part of the ski max(min( s−s 1 s 2 −s 1 , 1), 0), s = e e max rear part of the ski In the equations above, the constants H and D denote the snow hardness and snow damping parameters, respectively; L denotes the length of the ski segment and w the half of the width of the ski segment. The hypoplastic constitutive equation takes into account that if the penetration depths of the ski segments increase along the ski and the snow is increasingly compressed, the snow remains compressed. A reasonable assumption is, that the maximum penetration depth along the ski occurs at the ski segment centered below the ski binding 2, 3 . Therefore, we dived the ski into a loaded front part and an unloaded rear part 2 . In the loaded front part we set f(s) = 1 and in the unloaded rear part the reduction of f(s) due to hypoplasticity is given by equation 3 and the constants s 1 and s 2 . In summary, incorporation the hypoplastic constitutive equation into the ski-snow contact model offered the possibility to model the behaviour that during a carved turning maneuver in skiing the front part of the ski typically forms a snow groove and the rear part of the ski follows the groove 1-3 .
In the simulation of the turning maneuver, we modified the ski-snow contact forces to be twice differentiable, which is required when using gradient-based optimization. Specifically, we used the following smooth approximations in Table S1, where e 0 , D 0 , θ 0 and s 0 denote constants. In the smooth approximation of the scaling function f , we introduced the variables to assure that the scaling function f never exceeds 1.

S2 Results
Additional results are shown in Figs. S1 to S5 as well as Table S1 and S2.  Table S1. Smooth approximations used in the ski-snow contact model. Figure S1. Comparison of the optimized track of the skier in the turning simulation (solid lines) and the corresponding measurement data (dashed lines) in (a) as well as the the measured (gray) and optimized speed (yellow) of the skier (b) . Additionally, the turn radius of the center of mass of the skier (c) as well as the total ground reaction force, the ground reaction force acting on the outside ski (blue) and inside ski (red) are shown (d) .  4 and Harris et al. 5 . Joint moments were represented as internal joint moments and hip flexion, adduction and internal rotation, knee extension and ankle dorsiflexion moments were denoted as positive. Figure S4. Sensitivity study analyzing the effect on the joint angles (a), joint moments (b), muscle activation patterns (c) and ACL forces (d) of the right outer leg using 50 (n50), 75 (n75), 100 (n100) and 125 (n125) mesh points in the optimization. Figure S5. Sensitivity study analyzing the effect of the choice of the initial guess in the optimization on the joint angles (a), joint moments (b), muscle activation patterns (c) and ACL forces (d) of the right outer leg. The initial guesses were a data-independent schussing simulation (schussing) as well as turning simulation (turning), that was computed using a PD controller to track the experimental joint angles of the skier followed by a muscle redundancy solver.  Table S2. Results of the sensitivity study regarding the three terms (i.e., tracking error, muscle effort, regularization) in the objective function J. The nominal values of the weighting coefficient were w 1 = 1, w 2 = 10, w 3 = 1. In the sensitivity study, the number of nodes (n50, ..., n125), the initial guess (schussing, turning), the weighting coefficient (w2.5, ..., w40) of the muscle effort term (i.e., w 2 ) and the exponent of a in the muscle effort term (a 2 , a 3 , a 5 ) were varied.